Question: Simplify and expand the following expression: $ \dfrac{2a}{5a + 9}-\dfrac{a}{3a + 8} $
In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(5a + 9)(3a + 8)$ Multiply the first term by $\dfrac{3a + 8}{3a + 8}$ $ \begin{align*} \dfrac{2a}{5a + 9} \times \dfrac{3a + 8}{3a + 8} & = \dfrac{(2a)(3a + 8)}{(5a + 9)(3a + 8)} \\ & = \dfrac{6a^2 + 16a}{(5a + 9)(3a + 8)}\end{align*} $ Multiply the second term by $\dfrac{5a + 9}{5a + 9}$ $ \begin{align*} \dfrac{a}{3a + 8} \times \dfrac{5a + 9}{5a + 9} & = \dfrac{(a)(5a + 9)}{(3a + 8)(5a + 9)} \\ & = \dfrac{5a^2 + 9a}{(3a + 8)(5a + 9)}\end{align*} $ Now we have: $ = \dfrac{6a^2 + 16a}{(5a + 9)(3a + 8)} - \dfrac{5a^2 + 9a}{(3a + 8)(5a + 9)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{6a^2 + 16a - (5a^2 + 9a)}{(5a + 9)(3a + 8)} $ $ = \dfrac{6a^2 + 16a - 5a^2 - 9a}{(5a + 9)(3a + 8)} $ $ = \dfrac{a^2 + 7a}{(5a + 9)(3a + 8)}$ Expand the denominator: $ = \dfrac{a^2 + 7a}{15a^2 + 67a + 72}$